ℂ-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation

We investigate the concepts of linear convexity andℂ-convexity in complex Banach spaces. The main result is that anyℂ-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given aℂ-convex domainΩin the Banach spaceXand a pointp∉Ω, there is a complex hyperplane throughpthat does not intersectΩ. We also prove that linearly convex domains are holomorphically convex, and that Kergin interpolation can be performed on holomorphic mappings defined inℂ-convex domains.

Local uniform linear convexity with respect to the Kobayashi distance

We introduce the notion of local uniform linear convexity of bounded convex domains with respect to their Kobayashi distances.